Tracking device with deferred activation and propagation of passive tracks

ABSTRACT

A tracking device is configured to estimate a track for at least one possible target and is configured to receive incoming measurements and to process measurements and tracks. The tracking device includes a storage and a computational device. The tracking device is also configured to divide all measurements into a set of considered measurements and a set of unconsidered measurements for each passive track.

FIELD OF INVENTION

This invention relates to a tracking device configured to estimate atrack for at least one possible target and configured to receiveincoming measurements and to process measurements and tracks.Specifically, the invention relates to a tracking device in whichestimated tracks are maintained as active tracks or transferred to a setof passive tracks, deferring computations on the set of passive tracksuntil a handling criterion is fulfilled.

BACKGROUND OF THE INVENTION

Tracking of more than one target at the same time is an on-going fieldof development.

In practice a typical sensor is a radar sensor measuring range anddirection to possible targets. A lot of measurements are false, unwantedcomes from rain clouds, waves or other objects or simply due to noiseand of no interest to the user.

In most practical applications, the user expects an upper limit to thelatency through the system, being the time between a measurement is madeby a sensor and a track is reported on the user interface.

A plurality of implementations of tracking and tracking devices doexist, but they are limited either due to complexity or non-convergenceof calculations and thus results in high latency in the system using atracker or a tracking device.

In CHEE-YEE et al., “Efficient multiple hypothesis tracking by tracksegment graph”, Information Fusion, 2009. FUSION '09, 12^(th)International Conference on pp. 2177-2184, 6-9 Jul. 2009, tracking usinga track segment graph is discussed. In this reference, ambiguous tracksegments of multiple targets are maintained to generate a long term, oran extended duration, track hypothesis in case of ambiguity. Thus, thelong term track hypothesis is provided only when needed, for examplewhen the path of two targets almost overlap and make data associationdifficult. The long term track hypothesis is maintained to enablesolving of the ambiguity when further feature data on the targets areprovided, for example from infrequent video recordings of the targets.The long term track hypothesis is maintained to enable swapping ofactive tracks between targets when the ambiguity is resolved. Thisapproach is working only when further feature data are available and ismost useful when track swapping is the problem.

U.S. Pat. No. 5,414,643 discloses a tracking device in which two sets oftracks, a primary and a secondary, are maintained for each sensor. Theprimary tracks are the tracks which currently best represent the targetsin a cluster. The tracks in the secondary set of tracks having lessprobability than the tracks in the primary set. However, both primaryand secondary tracks are maintained actively increasing the requiredcomputational capacity.

Theoretical suggestions have been made on how to address trackingmultiple targets.

For example IEEE 2005, Intn'l Conf on Systems, Man & Cybernetics,Hawaii, October 2005, Bar-Shalom et al discuss “The Dimensionless ScoreFunction for Multiple Hypothesis Decision on Tracking”, hereinafterreferred to as Bar-Shalom 2005.

It is an objective to overcome limitations of the prior art.

SUMMARY

According to one aspect of the present invention a tracking device isprovided, the tracking device being configured to estimate a track forat least one possible target and configured to receive or obtainincoming measurements and to process measurements and tracks. Thetracking device is configured with a storage and a computational devicehaving a computational capacity and configured with an associationmodule configured to calculate an association between a measurement anda track. The tracking device further comprises an output moduleconfigured to output a sequence of track updates from an assignmentmodule maintaining a set of active tracks and, using the associationmodule to calculate associations between active tracks and the incomingmeasurements, the calculated associations containing possible trackupdates, deciding which track updates to keep in the set of activetracks and which track updates to add to a set of passive tracks; thecomputational capacity being configured to defer computations on the setof passive tracks until at least one passive track handling criterion isfulfilled. The computational device may furthermore be configured toactivate at least one passive track from the set of passive tracks andtransfer the at least one passive track from the set of passive tracksto the set of active tracks by at least one of the computations.

The passive track handling criterion may be a function of availablecomputational capacity and/or a function of track probability or tracklikelihood.

The association module may be configured to calculate an associationbetween a measurement and a track and, typically, a filter, such as aKalman filter or any other filters, such as filters of the same orsimilar class or type, such as an interacting multiple model estimator,such as possibly more advanced filters of the same or similar class ortype, may be used to calculate or estimate the likelihood of theassociation. Such filter types may be as described for example inBar-Shalom 2005.

The association module may use one or more gating mechanisms so thatassociations with zero associability can be calculated withsubstantially less computational resources than otherwise. Typically,filtering may be performed after a track and measurement pair passesthrough the gate(s).

The output module may be configured to output tracks or at least trackupdates to the user or for further processing. If only track updates aresent out, the user can concatenate the track updates to a full track.The user may only receive a subset of the active tracks.

The assignment module may estimate the best set of tracks to bepublished via the output module using the associability as calculated bythe association module to provide an assignment decision.

The assignment module may maintain a set of active tracks using theassociation module as a function of active tracks and the incomingmeasurements to calculate associations, containing possible trackupdates and deciding which track updates to keep in the set of activetracks and which track updates to add to a set of passive tracks.

The assignment module may thus, maintain a set of active tracks. Theassociation module may calculate associations between active tracks andthe incoming measurements and the calculated associations containpossible track updates. Based on the calculated associations, and thusthe possible track updates, the assignment module may decide which trackupdates to keep in the set of active tracks and which track updates toadd to a set of passive tracks; The assignment module may be, or may beimplemented as an assignment module as known from the prior art. Assuch, a known or existing assignment module may be reconfigured ormodified according to disclosures herein. An Implementation may be whereprovision is made to store otherwise discarded tracks and track updatesas passive tracks.

One advantage is that the establishment of a set of passive tracks mayenable implementation of or may allow the system to choose a simple andrelatively fast assignment module for maintaining the set of activetracks such as, but not limited to, a Nearest Neighbour class or type ofalgorithm to yield a low latency output of the tracking device. At thesame time it is able to use more complex implementations such as, butnot limited to, multi-dimensional or multi-hypothesis type algorithm oftracking.

By the distinction between active tracks and passive tracks and byestablishing and handling passive tracks, according to this disclosure,it is possible to bring back passive tracks into the set of activetracks, which active tracks are available to be maintained from theassignment module. This allows for a workable implementation that willimplement multi hypothesis tracking types of algorithms without the needto calculate all possible tracks within a time window; or without a needto simultaneously maintain several hypotheses.

It is an advantage of the present invention that by reducing the amountof calculations, the number of active tracks may be reduced compared toconventional MHT methods. The gained computational resources or capacitymay be used to allow a higher rate of incoming measurements ifconstrained to the same computational capacity.

Passive tracks, i.e. tracks added to the set of passive tracks by theassignment module, may be tracks which are not maintained, or notregularly maintained, and, thus, passive tracks may not be updated withincoming measurements, and the passive tracks are therefore, typically,not updated when new measurements or frames of measurements incoming atthe tracker.

It is an advantage of adding tracks to a set of passive tracks which arenot maintained, or not regularly maintained, in that the passive tracksmay be kept for a much longer time, before discarding or pruning ascomputational capacity or power is not required to keep the passivetracks. Computations on the set of passive tracks may be deferred orpostponed for more than 20 frames, such as up to 25 frames, such as upto 75 frames, such as up to 100 frames.

Typically, computations on the set of passive tracks are deferred untilat least one track handling criterion is fulfilled. The at least onetrack handling criterion may be a function of available computationalcapacity. Thus, computations may be performed when there is unused oradditional computational capacity. Furthermore, the at least one trackhandling criterion may be a function of track probability or tracklikelihood.

It is an advantage of the present invention that track fragmentation maybe reduced, thus, typically, in track fragmentation one track may belost, and a new track may be detected without realising that it is thesame target which has moved differently than predicted. However, withthe present invention, it may be possible to activate a passive trackwhich follows the target from the time a false decision on theassignment was taken, without having to maintain or update all passivetracks in real time, that is maintain or update passive tracks as ifthey were active tracks. This may in turn increase the latency of thesystem, however, it has been found that an acceptable latency may beobtained by the present invention, and particularly by performingcomputations in dependence of available computational capacity, thelatency can be kept at an acceptable level. In one or more embodiments,the tracking device may be configured to prioritize the computations onthe set of passive tracks based on a function of track likelihood ortrack probability. Thus, if a passive track has a track likelihood or atrack probability which is comparable with a track likelihood or a trackprobability of the active track, the computations on the set of passivetracks may be prioritized.

Even though computations have not been continued on a specific passivetrack, the track probability of the passive track may increase when thetrack probability or track likelihood of the competing active track isdecreasing.

The activation of a passive track, and likewise the passivation of anactive track may be performed by an activation module configured toactivate a passive track, respectively passivate an active track.

Besides reduction in algorithmic complexity from at least O(M^(N)), Mbeing the number of measurements per frame and N being the number offrames in the time window, the disclosed tracking device may have acomplexity as low as to O(N M log M) and therefore becomes moreefficient since it requires less computational resources and thus alsosaves computational resources, which are limited or expensive.

A person skilled in the art of working with tracking devices and/orradar systems will recognise that the field and literature of the fieldmay use different terms for the same features.

A person skilled in the art will appreciate that a measurement may bedata records possibly obtained from a sensor. A measurement may containinformation about the position of and possibly some other features, suchas radar intensity, corresponding to a possible target. A measurementmay originate from a true target, but many measurements may also befalse measurements, for instance originating from noise.

A measurement may also be an observation or basically a record of dataprovided.

A measurement may also just be a data item containing information aboutthe sensor, thus not what is actually sensed. For instance: If thesensor is a rotating radar, it may send dummy measurements of thecurrent antenna position. These dummy measurements have zeroassociability with any track.

For measurements or equivalents, a track is here defined as a sequenceof measurements, τ={m₁, m₂ . . . }, which, when connected, might be atrue series of measurements originating from the same target; and apossible target being a possible real world object being tracked.

For a track there may be a track update, which may be a track appendedor linked with a measurement creating a new track, τ′=τ∪{m}={m₁, m₂ . .. , m}.

A measurement may fit a track. How well the fit is, is described ortermed as assocability. Thus associability may be implemented by ascoring function for how well a measurement fits with a track. Thescoring function, and thus the associability, may be a number informingthe system how likely it is, or a measure of how likely it is, given thepossible track is a correct set of measurements for a target, that theextension of the track with a measurement is a correct set ofmeasurements for the target.

A filter may be used, where the so called continuation likelihood ratiois used for associability, such as a for example a Kalman filter or anyother similar filter. In the present disclosure, any likelihood forassociation between a track τ and a measurement m may be denoted L(τ,m).

The function of an association module or “associator” is to calculate,estimate or otherwise provide an association for a given possible trackand a measurement, where association is the associability between atrack and a measurement, and, if the associability is high enough, alsothe resulting track update. The associability may be a measure ofcorrelation between the track and the measurement.

The tracking device is furthermore configured to perform an assignment.The assignment may be a process of choosing which track updates to workon, and which to discard, after new measurements have been included.

A two-dimensional or 2-D assignment may start with what is considered tobe the best hypothesis of M possible tracks, and after a set of Nmeasurements have been received and M×N associabilities have beencalculated, then the 2-D assignment yields a new set of trackscontaining between M and M+N tracks, which is now considered to be thebest hypothesis.

An assignment module is a module performing assignment. The assignmentor assignment module may be Implementations of a 2-D assignment, but mayalso be a K-best hypotheses assignment. Multi-Dimensional Assignment(MDA), or even K-best MDA.

The tracking device may be configured to handle or process activetracks. The set of active tracks is the set of tracks that theassignment module updates, maintains or handles. When a 2-D assignmentalgorithm is implemented, this is an approximation to the besthypothesis. A person skilled in the art will appreciate otherimplementations. In a K-best assignment this may be the union of theK-best hypotheses, in an MDA implementation it may the whole of thetrack tree on which the MDA is calculating. In a K-best MDAimplementation it may the union of the K track trees that the K-best MDAuses.

A feature of operating on passive tracks and active tracks is toactivate tracks. This is the task of promoting a track from the set ofpassive tracks or passive container to the set of active tracks.

Different algorithms can be applied. Using the known MDA algorithms,such as Lagrangian relaxation, to estimate the best global hypothesisfor all tracks, passive and active, is one possible way to populate thebest hypothesis used for 2-D assignment MDA algorithms exist to find theK-best hypotheses when the assignment method is based on a K-bestmethod. A person skilled in the art may implement various tasks of thetracking device in a modular way, or alternatively implement thetracking device as a monolith or as a single unit. Likewise, a personskilled in the art will readily acknowledge the need to be able toprovide storage, means for moving data records around and makeprovisions for deleting useless data to avoid running out of storage.

The computational device may be a processing unit or another devicehaving a computational capacity, i.e. a computer processing capacity,and may comprise a CPU or any other processor having a given finalprocessing power or computational capacity. Each module as hereindiscussed may be implemented in the computational device, each modulebeing dedicated to performing tasks as allocated to this module. Themodules may be implemented either as separate modules or as one moduleperforming a plurality of tasks.

In one or more embodiments, the tracking device is further configured todivide all measurements into a set of considered measurements and a setof unconsidered measurements for each passive track. The tracking devicemay be configured with a propagator module configured to propagate aselected passive track using the association module to calculate anassociation as a function of an unconsidered measurement and theselected passive track; adding the possible track update of the selectedpassive track and the unconsidered measurement to the set of passivetracks, and marking the measurement considered for both the selectedpassive track and the possible track update.

The division of all measurements may be performed since the passivetracks are not associated with incoming measurements received orobtained after a passive track has been added to the set of passivetracks. With no knowledge of which measurements that might be associatedwith a passive track, it may be difficult to build the correct tracktree and a track might skip one or more measurements. For an activetrack, on the other hand, all measurements have been considered,depending on the mode of operation in the assignment module.

If the association module uses gating to optimize its calculations and atrack and a measurement is found to be outside the gating, the pair maystill be defined as considered, however, the pair maybe disregarded bygating.

One pre-requisite for a passive track to be activated may be that allmeasurements must have been considered. An implementation using a 2-Dassignment works better if this is enforced but it is not a requirementfor a working implementation.

The propagator module may allow for calculation of parts of a track treeindependently of the assignment module. The full track tree, which iscalculated as suggested in the prior art track oriented multi hypothesistracking (TOMHT), might still be calculated if computational capacity isavailable, but the propagator module might only calculate the partswhich have the most interest. Therefore implementations according todisclosures herein may save considerable computational capacity.

Furthermore, track propagation may be deferred to other computationalunits. An implementation may utilize multiple CPUs or other multipleprocessor configurations.

In one or more embodiments, a division of considered measurements andunconsidered measurements is performed by using an ordered identifier ofincoming measurements and storing the ordered identifier of the latestconsidered measurement on each passive track; and the track propagatoris configured to propagate a selected passive track according to theordered identifier, advancing the ordered identifier of the latestconsidered measurement of the selected passive track.

The measurements may be stored in a linear container or a list. Theincoming measurements may be appended to the container. The orderedidentifier may be an index or a timestamp indexing into this container.

By storing the ordered identifier of the latest considered measurementon each passive track, it becomes a straightforward implementation tofind the measurements with an identifier larger than the storedidentifier and then propagate the track by calculating associations tothe next measurements in the measurement container or list.

In one or more embodiments, passive tracks may be propagated as afunction of available computational capability. In some embodiments,passive tracks may be activated as a function of available computationalcapacity. The tracking device may be configured to propagate the passivetracks.

By lowering the prioritization of the propagation module and the trackactivation, the tracking device will primarily use computationalcapacity for associating incoming measurements and tracks and assigningtracks to the set of active tracks and the set of passive tracks,respectively. Thus, the computational capacity may primarily be used onthe association module and the assignment module. The prioritization ofthe propagation module may for example be lowered when the rate ofincoming measurements becomes high. In practice, this may be during abad weather situation, where a radar sensor might output many extrameasurements. Thus, the degree of multi hypothesis tracking can smoothlybe scaled up or down depending on availability of the computationalcapacity according to actual circumstances.

Furthermore, the tracking device may be configured to passivate anactive track and transfer the active track from the set of active tracksto the set of passive tracks. If the computational capacity istemporarily too low for even the assignment module to operate correctly,passivating active tracks to the set of passive tracks may be a way toovercome the starvation without deleting data.

Provisions of storage may be provided as needed and tracking device maybe configured with a measurement module configured with a measurementcontainer configured to store and process measurements within thetracking device.

Likewise, the tracking device may be configured with a pruning moduleconfigured to discard or delete unwanted or unimportant tracks.

To avoid that a set of passive track grows beyond capacity or thestorage is exceeded, provisions for a removing or pruning module isneeded. One option is to delete all passive tracks that have an estimateof probability that is under a threshold. Another option is to use asliding window; which is known from prior art TOMHT implementations.I.e. this is to eliminate branches in the track tree spanning more thana fixed time interval before the latest received or obtainedmeasurement, while only maintaining the branch in the tree having thehighest estimate of probability, or maintaining the part of the tracktree containing an active track.

In this disclosed implementation, the time interval may be largecompared to other TOMHT implementations, because the track tree is muchmore sparse.

In one or more embodiments, the tracking device may be configured toprioritize propagation of passive tracks based on a function of tracklikelihood, L(τ). Thus, the propagation of passive tracks may be basedon a function of track likelihood, L(τ).

A cumulative log-likelihood or negative track score l(τ) as defined inBar-Shalom 2005 may be used, or for example an equivalent tracklikelihood, L(τ)∝exp(−l(τ)), or simply L(τ)=exp(−l(τ)), may be used.

When no measurement is assigned to a track, but one was expected withprobability P_(D) (probability of detection), L(τ) may be multipliedwith 1−P_(D). In the case of a rotating radar, this happens when theantenna has scanned or passed the direction where the track is expectedto be for a given scan.

Applying notations used herein, the likelihood can be calculatedrecursively as

L(r∪{m})=L(τ)L(τ,m),

with L(τ,m) being the likelihood of association, or continuationlikelihood.

In one or more embodiments, the estimate of track probability P(t) maybe calculated as:

${P(\tau)} \simeq {P_{0}(\tau)} \equiv \frac{L(\tau)}{L_{0} + {\sum\limits_{\tau^{\prime} \in t}{L\left( \tau^{\prime} \right)}}}$

where t is the target for which r is a possible track.

L₀ is the likelihood of the target not being a real target. If thetarget is known to be real L₀=0.

In many cases L₀=1, but L₀ and L(τ) can freely be multiplied by aconstant so as to fit numerically within a machine representable range.

When the tracking device is configured to initiate tracks automatically,a possible new target with one track may be created for eachmeasurement. This so-called singleton track consisting of a singlemeasurement r=(m typically has a very low likelihood, i.e. L({m})<<L₀,such that P₀(τ)<<1. L({m}) is essentially the ratio of density of newtargets to density false measurements, when L₀=1.

The track probability estimate may be a basis estimate for one targethaving several possible tracks. The estimate may not address more thanone target conflicting and competing for the same measurement. Also,when the tracker is configured to make new tracks from measurements, theprobability of a new track being parallel with an existing track may beoverestimated.

In one or more embodiments, the estimate of track probability, P(t) iscalculated by an approximation

${P(\tau)} \approx {P_{eff}(\tau)} \equiv \frac{{L(\tau)}{X(\tau)}}{L_{0} + {\sum\limits_{\tau^{\prime} \in t}{{L\left( \tau^{\prime} \right)}{X\left( \tau^{\prime} \right)}}}}$where${X(\tau)} \equiv \ {\min\limits_{{{other}\mspace{14mu} {targets}},t}\ {X_{t}(\tau)}}$where${X_{t}(\tau)} \equiv {1 - {\sum\limits_{{\tau^{\prime} \in \; t},{{\tau\bigcap{\tau \; \prime}} \neq \varnothing}}{P_{0}\left( \tau^{\prime} \right)}}}$where${P_{0}(\tau)} \equiv \frac{L(\tau)}{L_{0} + {\sum\limits_{\tau^{\prime} \in t}{L\left( \tau^{\prime} \right)}}}$

where t is the target (6) for which r is a possible track (4).

The advantage of this estimate is that, when a track of a small targetfollowing a large target will have a small exclusion factor X, becausethe measurements are already occupied by the track of the large target.Therefore, the probability estimated of the correct track of the smalltarget veering away from the large target may be much higher using thisestimate.

In one or more embodiments, a passive track is activated into the set ofactive tracks when the track likelihood, L(τ), or track probability,P(τ) is over a predefined threshold.

The association module may calculate the track likelihood for each trackupdate.

It is a possibility to use the likelihood as a basis for prioritizationpropagation. A downside may occur since long living tracks may have avery large likelihood. Therefore, in some embodiments, only propagationsof passive tracks belonging to long living targets may be conducted.

In one or more alternative or additional embodiments, the trackingdevice may be configured to prioritize propagation of passive tracksbased on a function of an estimate of track probability, P(s).

Using an estimate of track probability for each passive track as a basisfor prioritizing track propagating may be advantageous, since the use ofcomputational capacity is optimized. This may be as computationalcapacity is used only on the most likely parts of the track tree. Thispart of the track tree may in turn change as new measurements arrive ortrack propagations are calculated.

In one or more embodiments of the tracking device, the assignment modulemay be based on one or more of the following assignment algorithms:

-   -   A. a Nearest Neighbor algorithm;    -   B. a 2-dimensional assignment algorithm;    -   C. an N-dimensional assignment algorithm;    -   D. a K-best multi hypothesis assignment; or    -   E. a K-best N-dimensional assignment.

One of the algorithms may be used at the time, but more algorithms maybe implemented for use at the same time. These assignment methods oralgorithms have shown to be working options and may be used forassignments. The algorithms may be implemented in the assignment module.Other assignment methods or combinations of the lists may beimplemented.

In one or more embodiments, the tracking device may be implemented usinga Nearest Neighbour or a 2-D assignment algorithm and activating tracksmay be performed by estimating the best hypothesis considering the setof active tracks and a subset of the set of passive tracks, andactivating tracks belonging to the hypothesis and passivating tracks notbelonging to the best hypothesis.

Such a method could be a Multi Dimension Assignment algorithm such asLagrangian relaxation, see for example Bar-Shalom 2005. This will,however, require that measurements can be arranged in some kind offrames.

In one or more embodiments, the best hypothesis among a set of tracks isestimated by repeatedly selecting the track with the highest estimate oftrack probability, P(t) and disregarding all conflicting tracks.

This method may not be as precise estimating the best hypothesis asLagrangian relaxation, but is advantageous since it does not require themeasurements to be arranged in frames.

Furthermore, the algorithm scales as O(T log T), where T is the numberof tracks considered. Furthermore, as tracks not estimated to belong tothe best hypotheses are not deleted, but kept in the set of passivetracks, any error may be corrected as further track propagations arecalculated and/or further measurements are received or obtained, as isillustrated in the detailed description.

In one or more embodiments using N-dimensional assignment, theactivating may be performed by building a set of tracks consisting of asubset of the active tracks and a subset of the set of passive tracksand iterating until the set is empty. Iteration may be performed as:move the track with the highest estimate of track probability to the setof active tracks; and move all tracks conflicting in any frame up to andincluding the N−1'th newest frame with any track in the selectedhypothesis to the set of passive track.

In one or more embodiments, the activating is performed by consideringone of the K hypotheses used in the assignment module and building a setof tracks consisting of a subset of the active tracks in the selectedhypothesis and a subset of the set of passive tracks and iterating untilthe set is empty: moving the track with the highest estimate of trackprobability to the set of active tracks in the selected hypothesis; andmoving all tracks conflicting with any track in the selected hypothesisto the set of passive tracks.

In one or more embodiments, the activating may be performed byconsidering one of the K hypotheses used in the assignment module andbuilding a set of tracks consisting of a subset of the active tracks inthe selected hypothesis and a subset of the set of passive tracks anditerating until the set is empty: moving the track with the highestestimate of track probability to the set of active tracks in theselected hypothesis; and moving all tracks conflicting in any frame upto and including the N−1^(th) newest and older frame with any track inthe selected hypothesis to the set of passive tracks.

It is understood that more tracks may be conflicting tracks, if they aredeemed to originate from the same target, or if they share a commonmeasurement, i.e. τ₁∩τ₂≠∅. Two tracks are conflicting in any frame up toand including the N−1^(th) newest frame if they are deemed to originatefrom the same target, or, if they share a measurement in any frame up toand including the N−1^(th) newest frame.

It is furthermore understood that a hypothesis is a set ofnon-conflicting tracks and that a best hypothesis may be the hypothesishaving the best combined likelihood.

According to a further aspect of the present invention, a method oftracking is provided, the method comprising processing a measurement anda track, which method of tracking encompasses:

-   -   receiving or obtaining a measurement;    -   associating the measurement with a track by calculating an        association between the measurement and the track;    -   assigning tracks by maintaining a set of active tracks by        associating and extracting possible track updates and deciding        which track updates to keep in the set of active tracks and        which track updates to add to a set of passive tracks, and defer        computations on the set of passive tracks until at least one        passive track handling criterion is fulfilled. The at least one        of the computations on the set of passive tracks may furthermore        activate a passive track from the set of passive tracks and        transfer the passive track from the set of passive tracks to the        set of active tracks;    -   outputting a track update.

Thus, an objective is achieved by a method performing acts as disclosedby modules or systems herein. Again, a person skilled in the art is notbound to strictly apply steps in a sequential order, and will realisethat actions may be implemented in parallel and sub-steps may beperformed as routine calls.

In one or more embodiments, the method of tracking further encompasses:

-   -   dividing all measurements into a set of considered measurements        and a set of unconsidered measurements for each passive track;        and    -   propagating, by computations on the set of passive tracks, a        selected passive track by associating by calculating an        association as a function of an unconsidered measurement and the        selected passive track and adding the possible track update of        the selected passive track and the unconsidered measurement to        the set of passive tracks, marking the measurement considered        for both the selected passive track and the possible track        update.

In some embodiments, the method of tracking further encompasses pruningor elimination of tracks.

In some embodiments, the method of tracking propagating of passivetracks is based on a function of track likelihood, L(τ), or on afunction of an estimate of track probability, P(τ) In some embodiments,the method of tracking assigning is based on one or more of thefollowing assignment implementations:

-   -   A. a Nearest Neighbor algorithm;    -   B. a 2-dimensional assignment algorithm;    -   C. an N-dimensional assignment algorithm;    -   D. a K-best multi hypothesis assignment; or    -   E. a K-best N-dimensional assignment.

In an embodiment the method of tracking propagating may be based on anestimate of track probability, P(τ), as discussed above.

An important feature in this implementation is that track likelihood maybe compared and used among tracks not having the same set of consideredmeasurements.

In a further aspect of the present invention, a method of tracking isprovided, the method of tracking comprising receiving or obtaining atleast one measurement from a sensor and processing the at least onemeasurement and at least one track, which method of tracking encompassesreceiving or obtaining the at least one measurement and associating theat least one measurement with the at least one track by calculating anassociation between the at least one measurement and the at least onetrack, and/or create a new track for example by track initialisation.Furthermore, the method may encompass calculating and using tracklikelihoods for assigning measurements to tracks while dividing allmeasurements into a set of considered measurements and a set ofunconsidered measurements for each track, which unconsideredmeasurements are deferred for later associating or assigning. The methodalso encompasses outputting a track update.

In the Bayesian probability theory, it is a basic assumption that alltracks must consider all received or obtained measurements for theformulas used in the described methods to be valid. It is an advantageof the present invention, that this may not be needed. The track treemay be partially updated, and updates may be deferred until later, butstill using the track likelihood or track score as though the wholetrack tree has been updated. For each track, one may use knowledge orinformation of which measurements have been considered and whichmeasurements are unconsidered to defer the update until later.

According to a still further aspect of the present invention, a trackingsystem comprising a sensor module and a tracking device is provided.Thus, the tracking device as herein described may be embedded in atracking system. The tracking system may be a radar system, or any othersensor system, comprising a sensor module and configured to providemeasurements to the tracking device. The sensor module may comprise atleast one or more sensors and a sensor control unit.

It is envisaged that all embodiments as described in connection withspecific aspects of the present invention, may be equally embodied witheach aspect.

BRIEF DESCRIPTION OF DRAWINGS

The embodiments will now be described more fully hereinafter withreference to the accompanying drawings, in which exemplary embodimentsare shown. The claimed Invention may, however, be embodied in differentforms and should not be construed as limited to the embodiments setforth herein. Embodiments of the invention will be described in thefigures, wherein:

FIG. 1 illustrates an embodiment of a tracking device with a set ofactive tracks and a set of passive tracks;

FIG. 2 illustrates an embodiment of a tracking device with a propagatormodule;

FIG. 3 illustrates division of considered measurements and unconsideredmeasurements;

FIG. 4 illustrates an embodiment of a tracking device with a propagatorand receiving module;

FIG. 5 illustrates a method of tracking;

FIG. 6 illustrates a method of tracking including propagating;

FIG. 7 illustrates definitions of targets, measurements, and tracks andtrack updates;

FIG. 8 illustrates definitions of targets, measurements, and tracks ofactive and passive sets of measurements; and

FIG. 9 to 15 illustrates a temporal evolution of tracking.

DETAILED DESCRIPTION OF THE DRAWINGS

Item No Tracking device 1 Measurement 2 Track 4 Track update 5 Target 6Sensor 8 Process 10 Computational device 12 Computational capacity 14Receiving module 20 Receiving time 22 Storage 23 Track likelihood 25Estimate of track probability 26 Conflicting tracks 27 Non-conflictingtracks 28 Association module 30 Associability 31 Association 32Assignment module 40 Maintaining 41 Keep 42 Add 43 Output Module 50Measurement container 70 Pruning Module 80 Hypothesis 90 Set of activetracks 100 Active track 102 Set of passive tracks 110 Division ofmeasurements 111 Passive track 112 Considered measurement 113Unconsidered measurement 114 Ordered identifier 115 Latest consideredmeasurement 116 Selected passive track 117 Advance 118 Activate 120Passivate 130 Propagator Module 150 Propagate 152 Receiving 200 Methodof tracking 222 Associating 300 Assigning 400 Outputting 500 Propagating600 Dividing 610 Pruning 700

FIG. 1 illustrates a tracking device 1 configured to estimate a track 4for at least one possible target 6 and configured to receive or obtainincoming measurements 2 and to process 10 measurements 2 and tracks 4.The tracking device 1 provides a track update 5 (not shown).

The measurement 2 may be an observation and may be obtained from asensor 8 (not shown).

The tracking device 1 is configured with a storage 23 (not shown) and acomputational device 12 (not shown) having a computational capacity 14(not shown). The computational device 12 may be a computer with one ormore processing units in in the form of central processing unit(s)(CPU), graphics processing unit(s) (GPU) or a field-programmable gatearray(s) (FPGA). It is understood that the computational device has alimited computational capacity 14.

The tracking device 1 may have an output module 50 configured to outputa sequence of track updates 5.

The tracking device 1 may have an association module 30 configured tocalculate an association 32 between a measurement 2 and a track 4.

In the illustrated embodiment the output module 50 receives the trackupdate 4 from an assignment module 40. The assignment module 40interacts with the association module 30. The assignment module 40provides a measurement 2 and an active track 102, possibly paired. Theassociation module 30 provides an association 32.

In this embodiment the assignment module 40 is configured to maintaining41 a set of active tracks 100 using the association module 30 as afunction of active tracks 102 and the incoming measurements 2 tocalculate or estimate associations 32, containing possible track updates5 and deciding which track updates 5 to keep 42 (not shown) or maintainin the set of active tracks 100 and which track updates 5 to add 43 toor to place in a set of passive tracks 110.

In this embodiment, the tracking device 1 is further configured toactivate 120 a passive track 112 from the set of passive tracks 110 andtransfer the passive track 112 from the set of passive tracks 110 to theset of active tracks 100.

FIG. 2 illustrates in continuation of FIG. 1 an embodiment where atracking device 1 configured with a propagator module 150 configured topropagate 152 a selected passive track 117 using the association module30. As will be illustrated in FIG. 3 the tracking device 1 is furtherconfigured to divide 111 (not shown) all measurements 2 into a set ofconsidered measurements 113 (not shown) and a set of unconsideredmeasurements 114 for each passive track 112.

The propagator module 150 is in this embodiment configured to propagate152 the selected passive track 117 using the association module 30 tocalculate an association 32 as a function of an unconsidered measurement114 and the selected passive track 117. The propagator module 150 thenadds the possible track update 5 of the selected passive track 117 andthe unconsidered measurement 114 to the set of passive tracks 110. Thepropagator module 150 then marks the measurement considered 113 for boththe selected passive track 117 and possible track update 5.

FIG. 3 illustrates division of measurement 111 between consideredmeasurements 113 and unconsidered measurements 114. For each passivetrack 112 associations 32 have been calculated between a passive track112 and each considered measurements 113. The remaining of themeasurements 2 are unconsidered 114 with respect to the passive track112.

In this embodiment division 111 of considered measurements 113 andunconsidered measurements 114 is performed by using an orderedidentifier 115 of incoming measurements 2 and storing the orderedidentifier 115 of the latest considered measurement 116 on each passivetrack 112.

This is understood in relation to FIG. 2 wherein the track propagator150 is configured to propagate 152 a selected passive track 117according to the ordered identifier 115 and advancing 118 the orderedidentifier 115 of the latest considered measurement 116 of the selectedpassive track 117.

FIG. 4 illustrates in continuation of the previous figures a trackingdevice 1 further configured with a receiving module 20.

The receiving module 20 may be configured to prepare measurements 20 forprocessing. The receiving module 20 may further be configured to assigna receiving time 22 (not shown) or generate an ordered identifier 115(not shown).

Furthermore the tracking devices 1 illustrated may be configured with astorage 23 (not shown), and a measurement container 70 (not shown).Finally, the tracking device may trivially be configured to eliminateobsolete data records using a pruning module 90 (not shown) orequivalent.

The shown tracking devices 1 may be obtained by implementing theoutlined functionalities in a different order or configuration.

In this embodiment, the propagation module 150 commands the associationmodule 30 to calculate an association 32 between the selected passivetrack 117 τ and an unconsidered measurement 114 such that thismeasurement becomes considered 113 (not shown).

The association 32 might contain a track update 5, τ′=T∪(m), which isinserted into the set of passive tracks 100. For all consideredmeasurements 117 the original r is also considered for new track τ′.

FIG. 5 illustrates a method of tracking 222. Features from the previousfigures are implemented or programmed based on the steps anddescriptions disclosed herein.

The method comprises receiving a measurement 2 (not shown), which may befrom a sensor 8 (not shown) and to process a measurement 2 and a track 4(not shown).

The method of tracking 222 encompasses receiving 200 a measurement 2.The method of tracking encompasses associating 300 a measurement 2 witha track 4 by calculating an association 32 (not shown) between ameasurement 2 and a track 4.

The method of tracking 222 includes assigning 400 tracks 4 bymaintaining 41 (not shown) a set of active tracks 100 (not shown) byassociating 300 and extracting possible track updates 5 (not shown) anddeciding which track updates 5 to keep 42 (not shown) in the set ofactive tracks 100 and which track updates 5 to add 43 (not shown) to aset of passive tracks 110 (not shown) and further encompassingfunctionality to activate 120 (not shown) a passive track 112 (notshown) from the set of passive tracks 110 and activate 120 (not shown)the passive track 112 from the set of passive tracks 110 to the set ofactive tracks 100.

Assigning 400 and associating 300 are interrelated in the same way asthe assignment module 40 and association module 30 illustrated in FIG.1.

The method of tracking 222 encompasses outputting 500 a track update 5.

FIG. 6 illustrates in continuation of FIG. 5 a method of tracking 222.Again reference is made to disclosures herein or previous figures. Inparticular reference is made to features described in FIG. 2.

The method of tracking 222 includes dividing 610 all measurements 2 (notshown) into a set of considered measurements 113 (not shown) and a setof unconsidered measurements 114 (not shown) for each passive track 112(not shown).

The method of tracking 222 includes propagating 600 a selected passivetrack 117 (not shown) by associating 300 by calculating an association32 (not shown) as a function of an unconsidered measurement 114 and theselected passive track 117 and adding the possible track update 5 (notshown) of the selected passive track 117 and the unconsideredmeasurement 114 to the set of passive tracks 110, marking themeasurement considered 113 for both the selected passive track 117 andthe possible track update 5.

For both embodiments of FIGS. 5 and 6 details about assigning 400 can befound in the description or summary regarding the assignment module 40.Likewise, details about associating 300 can be found in the descriptionor summary regarding the association module 30.

FIG. 7 illustrates track updates 5 of measurements 2 using an assignmentmodule based on a 2-D assignment. In this example two targets 6, targetI and target II, are tracked. Because 2-D assignment cannot havemultiple hypotheses there can only be one track 4 per target 6.

In the top figure two active tracks 102 and a single new incomingmeasurement 2 are shown.

In the bottom figure, two associations 32 between the two active tracks102 and the incoming measurement 2 are shown. The upper has the highestassociability 31 (not shown) and that track 4 is kept 42 (not shown). orremains, in the set of active tracks 100 (not shown).

FIG. 8 illustrates in the top part in continuation of FIG. 7 two activetracks 102 as the currently most likely representation of target I andtarget II, respectively.

In the top part, the set of active tracks 100 consists of two activetracks 102—one for each target 6—after processing the single incomingmeasurement 2 is shown.

In the bottom part, the set of passive tracks 110 is updated as a trackupdate 5 with the shown passive tracks 112. The update is performed bythe assignment module 40 (not shown) in the tracking device 1 (notshown) or by assigning 400 (not shown) in the method of tracking 222(not shown).

The FIGS. 9 to 15 describe a process of an embodiment in animplementation using a 2-D assignment implementation in the assignmentmodule 40 (not shown) exemplified with measurements originating from ascan-based radar sensor with a fixed scan period.

Each figure illustrates the situation with the logarithm of the tracklikelihood 25 (Log L) versus time (or radar period) plot (top) and thesame situation in a position versus time plot (bottom). All figuresillustrate the same features, which features are illustrated in FIG. 9(active tracks 102) and FIG. 10 (passive track 112) and remainself-explanatory here from.

FIG. 9 illustrates the initial tracking with one target 6 and one track4, τ₁={ . . . m₀, m₁}. Thus, the set of active tracks 100 is one activetrack 102, τ₁.

In FIG. 10 there are two incoming measurements 2, m₂ and m₃. Theassociation module 30 is used to calculate associations 32 (not shown)and two track updates 5, τ₂=τ₁∪{m₂}={ . . . m₀, m₁, m₂} and τ₃=τ₁∪{m₃}={. . . m₀, m₁, m₃}. Since τ₁ has a probability of detection P_(D)(τ₁),the track likelihood 25 L(τ₁) is multiplied with a factor of(1−P_(D)(τ₁)) as it has not been updated.

The assignment module 40 (not shown) or the step of assigning 400 (notshown) then considers tracks 4, τ₁, τ₂ and τ₃, and keeps τ₂ since it hasthe highest associability 31, here calculated as the log L(τ₂)−logL(τ₁), where L(τ₁) is the likelihood 25 of a track 4 τ. Thus the set ofactive tracks 100 (not shown) contains one active track (102), τ₂. τ₁and τ₃ are added 43 (not shown) to the set of passive tracks 110 (notshown) as passive tracks 112.

Continuing from FIGS. 9 and 10, in FIG. 11 one incoming measurement 2,m₄, is received or obtained. The association module 30 is used tocalculated the association 32 between the single active track 102, τ₂,and m₄. The associability 31 is too small because the distance betweenthe expected position of τ₂ and m₄ is too large, therefore there is notrack update 5 between τ₂ and m₄. Therefore the track likelihood 25L(τ₂) is multiplied with a factor of (1−P_(D)(τ₂)) leading to adecreased log L(τ₂). At this point the probability estimate of τ₃ isstill too low given the computational resources for τ₃ to be selectedfor propagation 600.

In FIG. 12 one incoming measurement 2, m₅, is received. The associationmodule 30 calculates the association 32 between the single active track102, τ₂, and m₅. The associability 31 is too small because the distancebetween the expected position of τ₂ and m₅ is too large, therefore thereis no track update 5 between τ₂ and m₄. Therefore, the track likelihood25 L(τ₂) is multiplied with a factor of (1−P_(D)(τ₂))

At this point the estimate of track probability 26 of τ₃ has grown dueto decrease in L(τ₂), and therefore there is now computational resourcesavailable for the passive track 112 τ₃ to be selected 117 forpropagation 600.

In FIG. 13 the passive track 112, τ₃, is propagated 152 by thepropagation module 150. The association module 30 calculates theassociation 32 between τ₃ and the unconsidered measurement 114, m₄. Theassociability 31 is large enough to generate a track update 5,τ₄=₃∪{m₄}={ . . . m₀, m₁, m₃, m₄}. Again the likelihood L(τ₃) is loweredby a factor (1−P_(D)(τ₃)) because track 4 τ₃ is not updated with ameasurement 2.

As this stage both the track likelihood 25 and estimate of trackprobability 26 of track 4 τ₂ exceed the track likelihood 25 and estimateof track probability 26 of track 4 τ₄.

In FIG. 14 the passive track 112, τ₄, is propagated 152 by thepropagation module 150. The association module 30 is used to calculatethe association 32 between τ₄ and the unconsidered measurement 114 m₅.The associability 31 is large enough to generate a track update 5,τ₅=τ₄∪{m₅}={ . . . m₀, m₁, m₃, m₄, m₅}. Again the likelihood L(τ₄) islowered by a factor (1−P_(D)(τ₄)) because track 4 τ₃ is not updated witha measurement 2.

Now both the track likelihood 25 and the estimate of track probability26 of track 4 τ₂ are lower than the track likelihood 25 and estimate oftrack probability 26 of track 4 τ₅.

Therefore and importantly as seen in FIG. 15, the passive track 112 τ₅is activated 120 and moved from the set of passive tracks 110 to the setof active tracks 100.

Because τ₂ and τ₅ are conflicting tracks 27 (not shown), and 2-Dassignment only can handle non-conflicting tracks 28 (not shown) In theset of active tracks 100, track 4 τ₂ is passivated 130 and moved to theset of passive tracks 110.

Although particular embodiments have been shown and described, it willbe understood that it is not intended to limit the claimed inventions tothe preferred embodiments, and it will be obvious to those skilled inthe art that various changes and modifications may be made withoutdeparting from the spirit and scope of the claimed inventions. Thespecification and drawings are, accordingly, to be regarded in anillustrative rather than restrictive sense. The claimed inventions areintended to cover alternatives, modifications, and equivalents.

1.-21. (canceled)
 22. A tracking device configured to estimate a trackfor at least one possible target and configured to receive incomingmeasurements and to process measurements and tracks, said trackingdevice comprising: a storage and a computational device having acomputational capacity and configured with: an association moduleconfigured to calculate an association between a measurement and atrack; an output module configured to output a sequence of track updatesfrom an assignment module; an assignment module configured to maintain aset of active tracks and, using the association module to calculateassociations between active tracks and the incoming measurements, thecalculated associations containing possible track updates, decidingwhich track updates to keep in the set of active tracks and which trackupdates to add to a set of passive tracks, the set of passive trackscomprises tracks which are not updated when new measurements arereceived at the tracker; the computational device being configured to:defer computations on the set of passive tracks until at least onepassive track handling criterion is fulfilled, activate at least onepassive track from the set of passive tracks and transfer the at leastone passive track from the set of passive tracks to the set of activetracks by at least one of the computations, the tracking device furtherconfigured to divide all measurements into a set of consideredmeasurements and a set of unconsidered set of measurements for eachpassive track; and comprising: a propagator module configured topropagate by the computations on the set of passive tracks, a selectedpassive track using the association module to calculate an associationas a function of an unconsidered measurement and the selected passivetrack; adding the possible track update of the selected passive trackand the unconsidered measurement to the set of passive tracks, andmarking the measurement considered for both the selected passive trackand the possible track update.
 23. The tracking device according toclaim 22, wherein the passive track handling criterion is a function ofavailable computational capacity and/or a function of track probabilityor track likelihood.
 24. The tracking device according to claim 22,wherein computations on the set of passive tracks are deferred orpostponed for up to 25 frames.
 25. The tracking device according toclaim 22, wherein a division of considered measurements and unconsideredmeasurements is performed by using an ordered identifier of incomingmeasurements and storing the ordered identifier of the latest consideredmeasurement on each passive track, and wherein the track propagatoris-configured to propagate a selected passive track according to theordered identifier, advancing the ordered identifier of the latestconsidered measurement of the selected passive track.
 26. The trackingdevice according to claim 22, configured to propagate passive tracks asa function of available computational capacity.
 27. The tracking deviceaccording to claim 22, configured to activate passive tracks as afunction of available computational capacity.
 28. The tracking deviceaccording to claim 22, configured to passivate an active track andtransfer the active track from the set of active tracks to the set ofpassive tracks.
 29. The tracking device according to claim 22,configured to prioritize the computations on the set of passive tracksbased on a function of track likelihood or track probability.
 30. Thetracking device according to claim 22, configured to prioritizepropagation of passive tracks based on a function of track likelihood,L(τ).
 31. The tracking device according to claim 22, configured toprioritize propagation of passive tracks based on a function of anestimate of track probability, P(τ).
 32. The tracking device accordingto claim 22, wherein the assignment module is based on one or more ofthe following assignments algorithms: A. a Nearest Neighbor algorithm;B. a 2-dimensional assignment algorithm; C. an N-dimensional assignmentalgorithm; D. a K-best multi hypothesis assignment; or E. a K-bestN-dimensional assignment.
 33. The tracking device according to claim 32,selections A or B, wherein activating tracks is performed by estimatingthe best hypothesis considering the set of active tracks and a subset ofthe set of passive tracks, and activating tracks belonging to thehypothesis and passivating tracks not belonging to the hypothesis. 34.The tracking device according claim 22, wherein the best hypothesisamong a set of tracks is estimated by repeatedly: selecting the trackwith the highest estimate of track probability, P(τ) disregarding allconflicting tracks.
 35. The tracking device according to claim 31,wherein the estimate of track probability P(τ) is estimated as:${P(\tau)} \simeq {P_{0}(\tau)} \equiv \frac{L(\tau)}{L_{0} + {\sum\limits_{\tau^{\prime} \in t}{L\left( \tau^{\prime} \right)}}}$where t is the target for which τ is an possible track.
 36. The trackingdevice according to claim 31, wherein the estimate of track probability,P(τ) is estimated by an approximation:${P(\tau)} \approx {P_{eff}(\tau)} \equiv \frac{{L(\tau)}{X(\tau)}}{L_{0} + {\sum\limits_{\tau^{\prime} \in t}{{L\left( \tau^{\prime} \right)}{X\left( \tau^{\prime} \right)}}}}$where:${X(\tau)} \equiv \ {\min\limits_{{{other}\mspace{14mu} {targets}},t}\ {X_{t}(\tau)}}$where:${X_{t}(\tau)} \equiv {1 - {\sum\limits_{{\tau^{\prime} \in \; t},{{\tau\bigcap{\tau \; \prime}} \neq \varnothing}}{P_{0}\left( \tau^{\prime} \right)}}}$where:${P_{0}(\tau)} \equiv \frac{L(\tau)}{L_{0} + {\sum\limits_{\tau^{\prime} \in t}{L\left( \tau^{\prime} \right)}}}$where t is the target for which τ is a possible track.
 37. A method oftracking comprising processing a measurement and a track, said method oftracking comprising: receiving a measurement; associating a measurementwith a track by calculating an association between a measurement and atrack; assigning tracks by maintaining a set of active tracks byassociating and extracting possible track updates and deciding whichtrack updates to keep in the set of active tracks and which trackupdates to add to a set of passive tracks, the set of passive trackscomprises tracks which are not updated when new measurements arereceived at the tracker; defer computations on the set of passive tracksuntil at least one passive track handling criterion is fulfilled,wherein at least one of the computations on the set of passive tracksactivate a passive track from the set of passive tracks and transfer thepassive track from the set of passive tracks to the set of activetracks; dividing all measurements into a set of considered measurementsand a set of unconsidered set of measurements for each passive track;propagating by the computations on the set of passive tracks, a selectedpassive track by calculating an association as a function of anunconsidered measurement and the selected passive track, adding thepossible track update of the selected passive track and the unconsideredmeasurement to the set of passive tracks, and marking the measurementconsidered for both the selected passive track and the possible trackupdate; and outputting a track update.
 38. The method of trackingaccording to claim 37, further comprising: dividing all measurementsinto a set of considered measurements and a set of unconsideredmeasurements for each passive track; and propagating a selected passivetrack by associating by calculating an association as a function of anunconsidered measurement and the selected passive track and adding thepossible track update of the selected passive track- and theunconsidered measurement to the set of passive tracks, marking themeasurement considered for both the selected passive track and thepossible track update.
 39. The method of tracking according to claim 37,wherein propagating of passive tracks is based on a function of tracklikelihood, L(τ), or on a function of an estimate of track probability,P(τ).
 40. The method of tracking according to claim 37, whereinassigning is based on one or more of the following assignmentsalgorithms: A. a Nearest Neighbor algorithm; B. a 2-dimensionalassignment algorithm; C. an N-dimensional assignment algorithm; D. aK-best multi hypothesis assignment; or E. a K-best N-dimensionalassignment.
 41. The method of tracking according to claim 39, whereinpropagating is based on an estimate of track probability P(τ) estimatedas:${P(\tau)} \simeq {P_{0}(\tau)} \equiv \frac{L(\tau)}{L_{0} + {\sum\limits_{\tau^{\prime} \in t}{L\left( \tau^{\prime} \right)}}}$where t is the target for which τ is an possible track; or estimated byan approximation:${P(\tau)} \approx {P_{eff}(\tau)} \equiv \frac{{L(\tau)}{X(\tau)}}{L_{0} + {\sum\limits_{\tau^{\prime} \in t}{{L\left( \tau^{\prime} \right)}{X\left( \tau^{\prime} \right)}}}}$where:${X(\tau)} \equiv \ {\min\limits_{{{other}\mspace{14mu} {targets}},t}\ {X_{t}(\tau)}}$where:${X_{t}(\tau)} \equiv {1 - {\sum\limits_{{\tau^{\prime} \in \; t},{{\tau\bigcap{\tau \; \prime}} \neq \varnothing}}{{{P_{0}\left( \tau^{\prime} \right)}.{where}}\text{:}}}}$${P_{0}(\tau)} \equiv \frac{L(\tau)}{L_{0} + {\sum\limits_{\tau^{\prime} \in t}{L\left( \tau^{\prime} \right)}}}$where t is the target for which τ is a possible track.